| L(s) = 1 | + 16·2-s + 20.8·3-s + 256·4-s + 846.·5-s + 333.·6-s − 1.83e3·7-s + 4.09e3·8-s − 1.92e4·9-s + 1.35e4·10-s + 4.00e3·11-s + 5.33e3·12-s + 1.88e4·13-s − 2.94e4·14-s + 1.76e4·15-s + 6.55e4·16-s + 4.77e5·17-s − 3.07e5·18-s − 8.06e5·19-s + 2.16e5·20-s − 3.83e4·21-s + 6.40e4·22-s − 1.03e6·23-s + 8.53e4·24-s − 1.23e6·25-s + 3.00e5·26-s − 8.11e5·27-s − 4.70e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.148·3-s + 0.5·4-s + 0.605·5-s + 0.105·6-s − 0.289·7-s + 0.353·8-s − 0.977·9-s + 0.428·10-s + 0.0824·11-s + 0.0742·12-s + 0.182·13-s − 0.204·14-s + 0.0899·15-s + 0.250·16-s + 1.38·17-s − 0.691·18-s − 1.41·19-s + 0.302·20-s − 0.0429·21-s + 0.0583·22-s − 0.773·23-s + 0.0525·24-s − 0.633·25-s + 0.129·26-s − 0.293·27-s − 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 151 | \( 1 - 5.19e8T \) |
| good | 3 | \( 1 - 20.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 846.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.00e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.88e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.77e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.06e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.45e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.87e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.79e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.04e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.29e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.06e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.869677797588439929482685507273, −8.668909408087492415139558777215, −7.80321673243975878832638251513, −6.34509699115304910040826119276, −5.91899028497875235052397761609, −4.75688992314434719309047357985, −3.52231441814238318520083613308, −2.62549321367583483172699154983, −1.53120639963740377615474776345, 0,
1.53120639963740377615474776345, 2.62549321367583483172699154983, 3.52231441814238318520083613308, 4.75688992314434719309047357985, 5.91899028497875235052397761609, 6.34509699115304910040826119276, 7.80321673243975878832638251513, 8.668909408087492415139558777215, 9.869677797588439929482685507273
